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This expression gives a very large number for Q2, indicating that the molecules can be distributed uniformly between the two "states" in many different ways Many other values of Q2 are possible, each one of which is associated with a particular nonuniform distribution of the molecules between the two halves of the container The ratio of a particular Q2 to the sum of all possible values is the probability of that particular distribution The connection established by Boltzmann between entropy S and Q is given by the equation: S = klnQ (542) where k is Boltzmann's constant, equal to R / N A Integration between states 1 and 2 yields:

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This value for the entropy change of the expansion process is the same as that given by Eq (514), the classical thermodynamic formula for ideal gases Equations (541) and (542) are the basis for relating thermodynamic properties to statistical mechanics (Sec 164)

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51 Prove that it is impossible for two lines representing reversible, adiabatic processes on a P V diagram to intersect (Hint: Assume that they do intersect, and complete the cycle with a line representing a reversible, isothermal process Show that performance of this cycle violates the second law) 52 A Carnot engine receives 250 kW of heat from a heat-source reservoir at 79815 K (525 C) and rejects heat to a heat-sink reservoir at 32315 K (50 C) What are the power developed and the heat rejected 53 The following heat engines produce power of 95 000 kW Determine in each case the rates at which heat is absorbed from the hot reservoir and discarded to the cold reservoir (a) A Carnot engine operates between heat reservoirs at 750 K and 300 K (b) A practical engine operates between the same heat reservoirs but with a thermal efficiency r/ = 035

Carnot-engine thermal efficiency for the same temperatures ( a ) What is the thermal efficiency of the plant ( b ) To what temperature must the heat-source reservoir be raised to increase the thermal efficiency of the plant to 35% Again 7 is 55% of the Carnot-engine value

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55 An egg, initially at rest, is dropped onto a concrete surface; it breaks Prove that the process is irreversible In modeling this process treat the egg as the system, and assume the passage of sufficient time for the egg to return to its initial temperature

56 Which is the more effective way to increase the thermal efficiency of a Carnot engine: to increase TH with Tc constant, or to decrease Tc with TH constant For a real engine, which would be the more practical way

57 Large quantities of liquefied natural gas (LNG) are shipped by ocean tanker At the unloading port provision is made for vaporization of the LNG so that it may be delivered to pipelines as gas The LNG arrives in the tanker at atmospheric pressure and 1137 K, and represents a possible heat sink for use as the cold reservoir of a heat engine For unloading of LNG as a vapor at the rate of 9000 m3 s-I, as measured at 29815 K (25 C) and 10133 bar, and assuming the availability of an adequate heat source at 30315 K (30 C), what is the maximum possible power obtainable and what is the rate of heat transfer from the heat source Assume that LNG at 29815 K (25 C) and 10133 bar is an ideal gas with the molar mass of 17 Also assume that the LNG vaporizes only, absorbing only its latent heat of 512 kJ kg-' at 1137 K 58 With respect to 1 kg of liquid water: ( a ) Initially at 27315 K (O0C), is heated to 37315 K (100 C) by contact with a heat it reservoir at 37315 K (100 C)What is the entropy change of the water Of the heat reservoir What is AStota1 ( b ) Initially at 27315 K (O C), it is first heated to 32315 K (50 C) by contact with a heat reservoir at 32315 K (50 C) and then to 37315 K (100 C) by contact with a reservoir at 37315 K (100 C) What is AStotal ( c ) Explain how the water might be heated from 27315 K (0 C) to 37315 K (100 C) SO that AStotal 0 = 59 A rigid vessel of 006 m3 volume contains an ideal gas, Cy = ( 5 / 2 ) R ,at 500 K and 1 bar ( a ) If heat in the amount of 15 kJ is transferred to the gas, determine its entropy change (b) If the vessel is fitted with a stirrer that is rotated by a shaft so that work in the amount J of 15 k is done on the gas, what is the entropy change of the gas if the process is adiabatic What is Astotal What is the irreversible feature of the process

510 An ideal gas, C p = ( 7 / 2 ) R ,is heated in a steady-flow heat exchanger from 34315 K to 46315 K (70 C to 190 C) by another stream of the same ideal gas which enters at 59315 K (320 C)The flow rates of the two streams are the same, and heat losses from the exchanger are negligible

CHAPTER 5 The Second Law of Thermodynamics (a) Calculate the molar entropy changes of the two gas streams for both parallel and countercurrent flow in the exchanger (b) What is ASbtalin each case (c) Repeat parts (a) and (b) for countercurrent flow if the heating stream enters at 47315 K (200 C)

511 For an ideal gas with constant heat capacities, show that: (a) For a temperature change from TI to T2, AS of the gas is greater when the change occurs at constant pressure than when it occurs at constant volume (b) For a pressure change from PI to P2, the sign of AS for an isothermal change is opposite that for a constant-volume change 512 For an ideal gas prove that:

513 A Carnot engine operates between two finite heat reservoirs of total heat capacity C& and CL (a) Develop an expression relating Tc to TH at any time (b) Determine an expression for the work obtained as a function of CL, C i , TH, and the initial temperatures TH, and Tco (c) What is the maximum work obtainable This corresponds to infinite time, when the reservoirs attain the same temperature In approaching this problem, use the differential form of Carnot's equation,

and a differential energy balance for the engine, dW-dQc-dQ~=0 Here, Qc and QH refer to the reservoirs

514 A Carnot engine operates between an infinite hot reservoir and ajnite cold reservoir of total heat capacity Ch (a) Determine an expression for the work obtained as a function of C i , TH(= constant), Tc, and the initial cold-reservoir temperature Tco (b) What is the maximum work obtainable This corresponds to infinite time, when Tc becomes equal to TH The approach to this problem is the same as for Pb 513 515 A heat engine operating in outer space may be assumed equivalent to a Carnot engine operating between reservoirs at temperatures TH and Tc The only way heat can be discarded from the engine is by radiation, the rate of which is given (approximately) by:

IQcl = kATc where k is a constant and A is the area of the radiator Prove that, for fixed power output I W I and for fixed temperature TH, the radiator area A is a minimum when the temperature ratio TcITH is 075